Thursday, February 28, 2013

In Robert
Frost’s famous poem, The Road Not Taken (sometimes mistakenly referred
to as “The Road Less Travelled”), a traveler is faced with the difficult choice
of which road he should follow.

This brings
to mind a classic logic puzzle, with mathematical implications.

Here is my
poetic interpretation of the puzzle.

THE ROAD
NOT TAKEN – Mr. Wagneezy Version

(Line 1 by
Robert Frost)

Two roads diverge in a yellow wood

One leads to certain death

The other leads to riches untold

I stop to catch my breath

I soon discover I have no clue

Exactly which road is which

I look to the right, and then to the left

My eyes begin to twitch

Suddenly two gnomes appear

From out of nearby briars

One of them is a truthful gnome

The other one is a liar

In looking at these gnomes, alas

I cannot tell the difference

Which one speaks truth?Which one speaks lies?

I fight the urge to wince

These seemingly identical gnomes

Both know which road to take

But instead of making it clear to me

They make it nearly opaque

The gnomes agree that one of them

Will answer a single question

Once I get the answer

They will end the conversation

I still can’t tell which one speaks truth

And which one is the liar

They smirk at me, these pesky gnomes

That came out of the briars

I must determine what to say

It is a daunting task

To get the gnomes to reveal the way

What question should I ask?

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Successfully
getting these gnomes to show us which road to choose will take careful
planning.Since we have no way of
knowing which gnome tells the truth and which gnome lies, we must be able to
come up with a question that both gnomes would answer the same way.

Obviously,
the direct approach (e.g. “Which road leads to untold riches?”) will not work,
because the truth-teller would point to one road while the liar would point to
the other road.

Therefore,
we must ask an indirect question – one that incorporates both the truth and the
lie.We can accomplish this by asking
one gnome to tell us which road the other
gnome would point us towards.(E.g.
“If I asked the other gnome which road leads to untold riches, which road would
he point to?”)

The logic
here is that the truth about a lie gives the same result as a lie about the
truth.

More
specifically, if we happen to talk to the truth-teller, he will point to the
wrong road because that is the road the liar would have pointed to.

On the other
hand, if we happen to talk to the liar, he will also point to the wrong road
because that is not the road the
truth-teller would have pointed to.

In either
case, the wrong road will be indicated, and we can choose the other road to
travel on.

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For movie
buffs – a version of this puzzle appeared in the movie Labyrinth
:

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Earlier, I
mentioned that this puzzle has mathematical implications.

Let’s
deconstruct this puzzle into a similar mathematical question.

I hope you
agree with me that telling the truth is positive,
and lying is negative.

Consider
mathematical operations that can be performed on two numbers.Suppose one of the numbers is positive, and
the other number is negative, but we DON’T KNOW WHICH IS WHICH.

Which
operations are guaranteed to give us results with the same sign, regardless of
which number is positive?

If we
arbitrarily choose ±2 and ±3 for our numbers and use them to explore each
operation, we get the following:

As we can
see, multiplication and division are the only operations that
fit the bill.

(Not
coincidentally, multiplication and division have the same priority in the Order of Operations.)

Therefore,
the logical argument

“The truth about a lie is equivalent to a lie about the
truth”

seems to
match up with the mathematical concepts

“A negative times a positive is equivalent to a positive
times a negative”

and

“A negative divided by a positive is equivalent to a
positive divided by a negative”.

By the way –
multiplication and division both exhibit the desired property because

1)Multiplying is the same thing as dividing by the
reciprocal.

2)Dividing is the same thing as multiplying by the
reciprocal.

and

3)A
number’s sign (positive or negative) is the same as the sign of the number’s
reciprocal.

Friday, February 8, 2013

The
differences between the students who fail in math and the students who get
decent grades – C and above – are generally clear cut, and easily identifiable
(unless there is a learning disability involved).Most of these differences aren’t even math
specific… failing students in any subject can generally improve their grades by
improving one or more of the following areas:

1)Get organized

2)Do (and turn in) assignments on time

3)Be willing/able to seek help when needed

4)Be willing to put in the time/effort necessary
to be successful

5)Practice/Prepare for tests

6)If the people you hang out with aren’t committed
to success, hang out with different people

However, I
recently found myself wondering about what sets the truly excellent math
students apart from the merely-great math students.I have a number of students earning a grade
of A or A- (at or above 90%).

Why do a
select few of these students consistently
earn grades near 100% (or above 100%, if extra credit is offered) in math
class?

In analyzing
my students, I have noticed a few traits that differentiate merely-great math
students from highly successful math students.

1.Merely-great math students tend to believe that
knowing a lot and being good at math is the most important component of good
test taking.

Highly successful math students understand that
knowing a lot and being good at math is actually the second most important component of test taking.The most important component is being able to
successfully communicate your knowledge and math excellence to your
teacher/professor.

There are many merely-great math students who turn
in tests containing problems with ambiguity in the solutions.Their brains might have been doing all the
right steps, but their work is suspect.Highly successful math students provide complete solutions that are clearly mathematically sound.As a result, they tend to receive higher
scores on their tests.

Laziness also plays a role in this.Many merely-great math students seem to
follow the philosophy, “When in doubt, show less work.”Most highly successful math students follow the
philosophy, “When in doubt, show more work.”

2.Merely-great math students don’t tend to place a
high value on neatness.

Highly successful math students tend to make
neatness a priority.

I can’t count how many times I’ve been grading work
and discovered that a student got the wrong answer due to a mistake caused by
poor handwriting.I’ve seen “4” turn
into “9”, “z” turn into “2”, “7” turn into “1” and a myriad of other sloppy
mistakes.These mistakes often prove to
be the difference between a 95% test and a 100% test.

3.Merely-great math students use lectures and
class time to learn what they need to know in order to succeed.

Highly successful math students also use lectures
and class time to learn what they need to know, but they tend to use this as a starting point in their learning
process.They are adept at looking in
textbooks for additional examples in order to enhance the material covered in a
lecture.They are also eager to explore
alternative methods and they have a desire to know WHY a particular method
works or doesn’t work.

4.Merely-great math students do their best to find
out what material will be covered on a test.(Some are even quite assertive in trying to get specific review topics
from their instructors.)They then review
this material diligently to make sure they can complete the test successfully.

Highly successful math students emphasize material
they know will be covered on a test when studying.However, they also attempt to review/practice
all of the other concepts from a chapter.This is helpful in (at least) two ways:

1.Since concepts are often interrelated, highly
successful students gain a deeper understanding of the “key concepts” by
broadening their review.

2.Teachers who put extra credit items on their
tests will often draw from these “other” concepts.When these extra credit test items show up,
highly successful students are ready for them.

5.Merely-great
math students sometimes put too much emphasis on being right at the expense of
focusing on what went wrong.

Highly successful math students attempt to learn as
much as possible from their mistakes so that they can avoid making the same
kind of mistakes in the future.

I can think
of a number of students who are experts at “nickel-and-diming” teachers out of
extra points.They submit work that is
good (but not great) and then have to verbally defend their work and try to
convince the teacher that they deserve 100% credit. They rejoice when their
arguments are fruitful and they receive a small increase in score… their
mission has been accomplished.Sadly,
these students tend to get in the habit of submitting less-than-stellar work
and find themselves arguing with their teachers a lot.

Highly successful math students, on the other hand, are
willing to admit when their work is less than ideal.They may argue the merits of their work with
their teacher in an attempt to get more points, but their chief concern is
learning how to produce stellar work in the future that will be above reproach.

The above
list of traits is not exhaustive – there are probably more traits of highly
successful math students that I’ve missed.(If you’d like to add to my list, leave a comment below.)

Also, the
above list does not come from a scientific study.It is simply an anecdotal summary of my own
personal observations.